Properties

Label 2-418-209.5-c1-0-15
Degree $2$
Conductor $418$
Sign $0.168 + 0.985i$
Analytic cond. $3.33774$
Root an. cond. $1.82695$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.615 − 0.788i)2-s + (1.44 − 1.39i)3-s + (−0.241 + 0.970i)4-s + (1.24 − 0.779i)5-s + (−1.98 − 0.278i)6-s + (0.930 + 0.414i)7-s + (0.913 − 0.406i)8-s + (0.0352 − 1.00i)9-s + (−1.38 − 0.502i)10-s + (2.19 + 2.48i)11-s + (1.00 + 1.73i)12-s + (4.07 − 2.16i)13-s + (−0.246 − 0.988i)14-s + (0.712 − 2.85i)15-s + (−0.882 − 0.469i)16-s + (−0.137 − 3.93i)17-s + ⋯
L(s)  = 1  + (−0.435 − 0.557i)2-s + (0.831 − 0.803i)3-s + (−0.120 + 0.485i)4-s + (0.557 − 0.348i)5-s + (−0.809 − 0.113i)6-s + (0.351 + 0.156i)7-s + (0.322 − 0.143i)8-s + (0.0117 − 0.336i)9-s + (−0.436 − 0.159i)10-s + (0.662 + 0.748i)11-s + (0.289 + 0.500i)12-s + (1.12 − 0.600i)13-s + (−0.0658 − 0.264i)14-s + (0.183 − 0.737i)15-s + (−0.220 − 0.117i)16-s + (−0.0333 − 0.953i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.168 + 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.168 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(418\)    =    \(2 \cdot 11 \cdot 19\)
Sign: $0.168 + 0.985i$
Analytic conductor: \(3.33774\)
Root analytic conductor: \(1.82695\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{418} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 418,\ (\ :1/2),\ 0.168 + 0.985i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.27247 - 1.07341i\)
\(L(\frac12)\) \(\approx\) \(1.27247 - 1.07341i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.615 + 0.788i)T \)
11 \( 1 + (-2.19 - 2.48i)T \)
19 \( 1 + (3.36 + 2.76i)T \)
good3 \( 1 + (-1.44 + 1.39i)T + (0.104 - 2.99i)T^{2} \)
5 \( 1 + (-1.24 + 0.779i)T + (2.19 - 4.49i)T^{2} \)
7 \( 1 + (-0.930 - 0.414i)T + (4.68 + 5.20i)T^{2} \)
13 \( 1 + (-4.07 + 2.16i)T + (7.26 - 10.7i)T^{2} \)
17 \( 1 + (0.137 + 3.93i)T + (-16.9 + 1.18i)T^{2} \)
23 \( 1 + (4.68 - 3.93i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (4.36 - 1.25i)T + (24.5 - 15.3i)T^{2} \)
31 \( 1 + (-4.79 + 5.32i)T + (-3.24 - 30.8i)T^{2} \)
37 \( 1 + (3.55 - 2.58i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (0.264 - 0.255i)T + (1.43 - 40.9i)T^{2} \)
43 \( 1 + (7.84 + 6.58i)T + (7.46 + 42.3i)T^{2} \)
47 \( 1 + (-6.29 - 9.33i)T + (-17.6 + 43.5i)T^{2} \)
53 \( 1 + (8.91 + 5.57i)T + (23.2 + 47.6i)T^{2} \)
59 \( 1 + (0.405 - 0.600i)T + (-22.1 - 54.7i)T^{2} \)
61 \( 1 + (0.707 + 1.75i)T + (-43.8 + 42.3i)T^{2} \)
67 \( 1 + (-13.2 - 4.82i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (8.46 - 5.28i)T + (31.1 - 63.8i)T^{2} \)
73 \( 1 + (-2.68 - 5.50i)T + (-44.9 + 57.5i)T^{2} \)
79 \( 1 + (-13.7 + 1.93i)T + (75.9 - 21.7i)T^{2} \)
83 \( 1 + (3.68 - 0.783i)T + (75.8 - 33.7i)T^{2} \)
89 \( 1 + (0.234 + 1.32i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (-8.28 - 10.5i)T + (-23.4 + 94.1i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.08726885994212120024353105906, −9.877866620396946745497002377969, −9.144616734066293055737685960317, −8.351233984318477182499471368755, −7.58319879243026712022413817114, −6.51876376093209280464638612058, −5.13477684823876635795489696446, −3.68588677072575985406987220120, −2.30300805066124144171493840245, −1.43331425836737268826877513085, 1.80679035074770291942890950123, 3.54396405259206119905005819822, 4.37287468289665637724338416713, 6.07722171412294207684785548566, 6.47954588530668769110749969556, 8.237197034953332788977037223786, 8.526008213212627254197366700953, 9.479679503750894197389150384722, 10.33981169026074064813275373784, 10.95403421973797537025430307406

Graph of the $Z$-function along the critical line